"... a book that is a marvelous first step for the person wanting a rigorous development of stochastic calculus, as well as its application to derivative pricing. By focusing solely on Brownian motion, the reader is able to develop an intuition and a feel for how to go about solving problems as well as deriving results." --- Mark A. Cassano (see the full review from the Journal of Finance)
"The main results are reinforced with simple special cases, and only when the intuitive foundations are laid does the author resort to the formalism of probability.... This is one of the most interesting and easiest reads in the discipline; a gem of a book." --- D. L. McLeish in Short Book Reviews
"...the results are presented carefully and thoroughly, and I expect that readers will find that this combination of a careful development of stochastic calculus with many details and examples is very useful and will enable them to apply the whole theory confidently." --- Martin Schweizer (Berlin) from the review in Zentralblatt fur Mathematik: (0962.60001)
"I thoroughly enjoyed reading this book. The author is to be complimented for his efforts in providing many useful insights behind the various theories. It is a superb introduction to stochastic calculus and Brownian motion." --- Elias Shiu (from the review in JASA)
How it Started ...
This book was developed for my Wharton class "Stochastic Calculus and Financial Applications (Statistics 955).
The participants in this class are well-prepared highly-motivated students who are typically in the second or third year Ph.d. program in finance, economics, statistics or mathematics.
The goal of the course is to offer serious professional training in stochastic calculus for people who expect to spend a lifetime engaging quantitative models.
The book (and the course) are mathematically rigorous, but if you look through the book you will see that it is not stuffy. The book was designed to enable students to do serious work with a minimum of overhead.
The book is primarily about the core theory of stochastic calculus, but it focuses on those parts of the theory that have really proved that they can "pay the rent" in practical applications. The intention is also to coach people toward honest mastery. This means that one must be selective in the topics that are treated, and one must engage those topics to some depth.
Students who work through the book will have stochastic calculus as part of their active --- not passive --- toolkit.
1. Random walks and first step analysis
2. First martingale steps
3. Brownian motion
4. Martingales: The next steps
5. Richness of paths
6. Itô integration
7. Localization and Itô's integral
8. Itô's formula
9. Stochastic differential equations
10. Arbitrage and SDEs
11. The diffusion equation
12. Representation theorems
13. Girsanov theory
14. Arbitrage and martingales
15. The Feynman-Kac connection
Appendix I: Mathematical Tools
Appendix II: Comments and Credits
If you promise not to look a gift horse in the mouth, you can have access to the solutions to the problems in Stochastic Calculus and Financial Applications.
Or, for the moment, you can at least a good selection of them, but you have to keep something strange in mind.
These solutions sets do not have the same numbering as the problems in the text, so you will have to look up the solution you seek without the benefit of any correspondence between the problem number and the solution number. That is not such a big price, is it?
With that warning, please let me know if you find a bug in any of the solutions. These will be part of some future edition (2010, or later?). For the moment, I will just post the solutions to the first eight chapters.
- Solutions Chapter 1 and Chapter 2
- Solutions Chapter 3 and Chapter 4
- Solutions Chapter 5 and Chapter 6
- Solutions Chapter 7 and Chapter 8
You can read more about the book on the Amazon landing page, or explore the text in detail with Search Inside for "Stochastic Calculus and Financial Applications ."
SCFA is also part of the new Amazon electronic up-grade program where book purchasers get web access to a PDF of the book that they can read and mark-up from any place on the planet (See details).
Eventually I plan to provide links here to the index and to provide complete solutions for some NEW problems. Over time I would like to make this more like a community page and less like a publisher's flyer. I've made a little progress in that direction with the Cauchy-Schwarz home page, but it takes time. In the meanwhile, I am thinking about a problem book on Brownian motion. This book would also have problems that are directed toward stochastic calculus.
Merton and Scholes received their Bank of Sweden Prizes almost ten years ago, and it is this work more than any other that has created the stimulus for the study of stochastic calculus.
I have been teaching closely related material for all of those years. Still, only recently did I come to appreciate how informative these essays these are. Give them another look. You may be surprised to see how "they" have changed over the years.
Robert C. Merton, Lecture to the memory of Alfred Nobel (December 9, 1997) "Applications of Option-Pricing Theory: Twenty-Five Years Later."
Myron Scholes, Lecture to the memory of Alfred Nobel (December 10, 1997) "Derivatives in a Dynamic Environment."
There are two 2006 books that I think that everyone with an interest in financial modeling should read.
Donald MacKenzie, An Engine, Not a Camera: How Financial Models Shape Markets (MIT Press, 2006)
Mark Rubinstein, A History of the Theory of Investments: My Annotated Bibliography, (Wiley Finance, 2006)
I will eventually write some of my impressions from these books. Just as a teaser, I might note that Donald MacKenzie is a prize-winning sociologist, so his book is not your usual finance book.
It goes without saying that Rubenstein's book is unique. A grandmaster of the field looks at its most important papers and puts into clear prose what he sees as their main contributions. You could have sold me this book one page at a time for several bucks per page. There is not a ton of stochastic calculus in these books, but there certainly are some interesting connections that help explain how stochastic calculus found its place in the world.